(2+1)-dimensional Chern-Simons gravity as a Dirac square root

For simple enough spatial topologies, at least four approaches to (2+1)-dimensional quantum gravity have been proposed: Wheeler-DeWitt quantization, canonical quantization in Arnowitt-Deser-Misner (ADM) variables on reduced phase space, Chern-Simons quantization, and quantization in terms of Ashtekar-Rovelli-Smolin loop variables. An important problem is to understand the relationships among these approaches. By explicitly constructing the transformation between the Chern-Simons and ADM Hilbert spaces, we show here that Chern-Simons quantization naturally gives rise to spinorial wave functions on superspace, whose time evolution is governed by a Dirac equation. Chern-Simons quantum gravity can therefore be interpreted as the Dirac square root of the Wheeler-DeWitt equation.